Method and system for processing multidimensional data

ABSTRACT

A method and system for processing multidimensional data. A method of forming multiresolution representations of data includes partitioning the data in a first dimension at a first rate, and partitioning the data in a second dimension at a second rate, wherein the first rate is not equal to the second rate. In one embodiment, the first rate is set to one, and the second rate is computed as the ratio of a smoothness estimate of the first dimension data and a smoothness estimate of the second dimension data. The partitioned data may be processed further through data compression, noise removal, or similar data processing methods.

RELATED APPLICATION

This application is a continuation under 37 C.F.R. 1.53(b) of U.S.application Ser. No. 09/615,445 filed Jul. 13, 2000, which claimedpriority under 35 U.S.C. 119(e) from U.S. Provisional Application Ser.No. 60/143,784 filed Jul. 14, 1999, which applications are incorporatedherein by reference and made a part hereof.

GOVERNMENT SUPPORT

This invention was made with U.S. Government support under Navy, GrantNo. N00014-91-J-1152. The Government has certain rights in theinvention.

FIELD

The present invention relates to processing data, and more particularlyto processing multidimensional data.

BACKGROUND

Many multidimensional data processing algorithms are based onmultiresolution decompositions. These algorithms include, for example,compression algorithms, noise removal algorithms, and algorithms for thereconstruction of images. The more efficiently these algorithms operate,the better the modem communications and information processing systemsin which they are embedded operate. For example, efficient compressionalgorithms permit fast transmission of information in communicationsystems. Without efficient compression algorithms, multidimensional datarequires an unacceptable amount of bandwidth for transmission and anunacceptable amount of storage for archiving.

Consider, for example, a medical image, such as a mammographic screeningimage, which may be represented by four six-thousand pixel bysix-thousand pixel arrays. A mammographic screening image consists offour images, two images for each of two breasts. Of the two imagesassociated with each breast, one image is a top image and one image is aside image. A pixel is a “picture element,” which is an elementary unitof information contained in an image and is typically represented by anintensity level. If each pixel in the mammographic screening image isrepresented by sixteen bits, then each pixel may be encoded at one of65,536 possible intensity levels. To transmit the mammographic screeningimage without compression, 2.3 billion bits must be sent over acommunication link. A typical telephone line is capable of transmittingabout 56,000 bits per second, so transmission of a mammographicscreening image would require more than ten hours. A ten hourtransmission time is unacceptable for transmitting a mammographicscreening image, so image compression processing is used to reduce thetransmission time.

Prior to processing multidimensional data using some compressionmethods, such as wavelet compression, the multidimensional data isapproximated at several resolution levels. For example, two-dimensionalimage data is initially divided into two rows and two columns. Each rowand each column is subsequently divided into two rows and two columns.FIG. 1 is an illustration of a sequence of images 100, including images101, 103, 105, and 107, of multidimensional data partitioned into rowsand columns. The images 101, 103, 105, and 107 illustrate partitioning afirst dimensions 109 (rows) and a second dimension 111 (columns) at arate of one. Image 101 is partitioned in the first dimension 109 and thesecond dimension 111. Each of the partitions in image 101 is partitionedor divided to form image 103. Each of the partitions in image 103 arepartitioned or divided to form image 105. And each of the partitions inimage 105 are partitioned or divided to form image 107. The partitioningor subdividing of rows and columns continues until an acceptableresolution level is achieved. An acceptable resolution level is a levelat which data can be compressed, transmitted, and decompressed, suchthat the decompressed data includes the information contained in theoriginal data required by a viewer of the received data. For example, inthe mammographic screening example described above, the decompresseddata must contain enough information related to a cancerous tumor toallow a radiologist to identify the cancerous tumor by viewing themammographic screening images reconstructed from the compressed data.

Isotropic decomposition is one type of decomposition used in somemulti-dimensional data processing algorithms. To perform isotropicdecomposition, one begins with a function φ of one variable such thatthe set{φ(x−j)|jεZ}forms a Riesz basis for the span of these functions. Assume that φsatisfies the rewrite rule $\begin{matrix}{{\phi(x)} = {\sum\limits_{j}{a_{j}{\phi\left( {x - j} \right)}}}} & (1)\end{matrix}$for a finite set of coefficients a_(j). Let S_(k) be the space of allfunctions$S_{k}:=\left\{ {\sum\limits_{j}{c_{j}{\phi\left( {{2^{k}x} - j} \right)}}} \middle| {c_{j} \in R} \right\}$and choose a bounded projection P_(k) from L_(p)(R) to S_(k). Undercertain conditions (see Daubechies) any ƒεL_(p)(R) can be re-written as$f = {{\lim\limits_{k->\infty}P_{k}} = {{P_{0}f} + {\sum\limits_{k = 1}^{\infty}\left( {{P_{k}f} - {P_{k - 1}f}} \right)}}}$where, because of the rewrite rule (1), P_(k)ƒ−P_(k−1)ƒ is in S_(k).Thus, since P₀ƒεS₀,$f = {{{P_{0}f} + {\sum\limits_{k = 1}^{\infty}\left( {{P_{k}f} - {P_{k - 1}f}} \right)}}\quad = {{\sum\limits_{j \in Z}{d_{j}{\phi\left( {\cdot {- j}} \right)}}} + {\sum\limits_{k = 1}^{\infty}{\sum\limits_{j \in Z}{d_{j,k}{{\phi\left( {2^{k} \cdot {- j}} \right)}.}}}}}}$For suitable functions φ and special projectors P_(k), one can find afunction ψ, associated with φ, such that${{P_{k}f} - {P_{k - 1}f}} = {\sum\limits_{j \in Z}{c_{j,{k - 1}}{\psi\left( {2^{k - 1} \cdot {- j}} \right)}}}$(note the new scaling −2^(k−1) instead of 2^(k)) so that$f = {{\sum\limits_{j \in Z}{d_{j}{\phi\left( {\cdot {- j}} \right)}}} + {\sum\limits_{k = 0}^{\infty}{\sum\limits_{j \in Z}{c_{j,k}{{\psi\left( {2^{k} \cdot {- j}} \right)}.}}}}}$

For a function ƒ: R^(d)→R a similar decomposition holds. Define a set Ψof 2^(d)−1 functions defined for x=(x₁, . . . , x_(d))εR^(d) by$\Psi:={\left\{ {\left. {\prod\limits_{i = 1}^{d}{v_{i}\left( x_{i} \right)}} \middle| v_{i} \right. = {{\phi\quad{or}\quad v_{i}} = \psi}} \right\}\backslash\left\{ {\prod\limits_{i = 1}^{d}{\phi\left( x_{i} \right)}} \right\}}$together with the function${\Phi(x)} = {\prod\limits_{i = 1}^{d}{{\phi\left( x_{i} \right)}.}}$Then, under suitable conditions, any ƒ in L_(p)(R^(d)) can be written as$f = {{\sum\limits_{j \in Z^{d}}{d_{j}{\Phi\left( {\cdot {- j}} \right)}}} + {\sum\limits_{k = 0}^{\infty}{\sum\limits_{j \in Z^{d}}{\sum\limits_{\psi \in \Psi}{c_{j,k}{{\psi\left( {2^{k} \cdot {- j}} \right)}.}}}}}}$Note that, since x=(x₁, . . . , x_(d)) and the multi-index j=(j₁, . . ., j_(d)),ψ(2^(k) x−j)=ψ(2^(k) x ₁ −j ₁, . . . , 2^(k) x _(d) −j _(d)),i.e., each of the components x_(i) of x has been scaled by the sameamount, 2^(k).

One disadvantage of isotropic decomposition is that not all data isisotropic, and anisotropic multidimensional data is not efficientlyprocessed by algorithms based on isotropic decomposition.

For these and other reasons there is a need for the present invention.

SUMMARY

According to one aspect of the present invention, a method is describedfor forming multi-resolution representations of data. The methodincludes the operations of partitioning the data in a first dimension ata first rate, and partitioning the data in a second dimension at asecond rate, wherein the first rate is not equal to the second rate.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of a sequence of prior art images ofmultidimensional data being partitioned into rows and columns;

FIG. 2 is an illustration of a sequence of images of multidimensionaldata being partitioned according to the present invention; and

FIG. 3 is a block diagram of one embodiment of a system including acomputer-readable medium having computer-executable instructions forperforming a method according to the present invention.

DESCRIPTION

In the following detailed description of exemplary embodiments of theinvention, reference is made to the accompanying drawings which form apart hereof, and in which is shown by way of illustration specificexemplary embodiments in which the invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the invention, and it is to be understood thatother embodiments may be utilized and that logical, mechanical,electrical and other changes may be made without departing from thespirit or scope of the present invention. The following detaileddescription is, therefore, not to be taken in a limiting sense, and thescope of the present invention is defined only by the appended claims.

Some portions of the detailed descriptions which follow are presented interms of algorithms and symbolic representations of operations on databits within a computer memory. These algorithmic descriptions andrepresentations are the means used by those skilled in the dataprocessing arts to most effectively convey the substance of their workto others skilled in the art. An algorithm is here, and generally,conceived to be a self-consistent sequence of operations leading to adesired result. The operations are those requiring physicalmanipulations of physical quantities. Usually, though not necessarily,these quantities take the form of electrical or magnetic signals capableof being stored, transferred, combined, compared, and otherwisemanipulated. It has proven convenient at times, principally for reasonsof common usage, to refer to these signals as bits, values, elements,symbols, characters, terms, numbers, or the like. It should be borne inmind, however, that all of these and similar terms are to be associatedwith the appropriate physical quantities and are merely convenientlabels applied to these quantities. Unless specifically stated otherwiseas apparent from the following discussions, it is appreciated thatthroughout the present invention, discussions utilizing terms such as“processing” or “computing” or “calculating” or “determining” or“displaying” or the like, refer to the action and processes of acomputer system, or similar electronic computing device, thatmanipulates and transforms data represented as physical (electronic)quantities within the computer system's registers and memories intoother data similarly represented as physical quantities within thecomputer system memories or registers or other such information storage,transmission or display devices. As used herein, the term image isdefined to be a collection of data, and an image may be displayed by adisplay device, such as a cathode ray tube, a liquid crystal display, orsimilar device.

FIG. 2 is an illustration of a sequence of images 201-204 beingrepeatedly partitioned according to the present invention. Each of theimages in the sequence of images 201-204 is a representation ofmultidimensional data. In the illustration shown in FIG. 2, themultidimensional data is two-dimensional data, however, the presentinvention is not limited to processing two-dimensional data. Theexemplary embodiments of the present invention described in theapplication are described in terms of two-dimensional data becausetwo-dimensional data examples are most easily illustrated andunderstood. Those skilled in the art will recognize that the presentinvention is applicable to data having two or more dimensions.

The sequence of images 201-204 may represent a variety of data types.For example, in one embodiment of the present invention, the sequence ofimages 201-204 represents X-ray images. In an alternate embodiment, thesequence of images 201-204 represents magnetic resonance imaging (MRI)images. In another alternate embodiment, the sequence of images 201-204represents positron emission tomography (PET) images. In anotheralternate embodiment, the sequence of images 201-204 representsmulti-spectral images. In still another alternate embodiment, thesequence of images 201-204 represents video images in which one of thedimensions is time.

Each of the types of images described above, along with other types ofimages not described above, but which are also suitable for use inconnection with the present invention, share a common characteristicwhich is that the smoothness of the data is anisotropic. Data smoothnessis anisotropic when the data smoothness depends on the dimension of thedata in which the smoothness is being estimated. For example, fortwo-dimensional data having a first dimension and a second dimension,data having a smoothness estimate of two in the first dimension and asmoothness estimate of three in the second dimension is anisotropic.Examples of data that is anisotropic include hyperspectral image dataand video data. Hyperspectral image data has the same smoothness in eachof the two spatial directions in a fixed frequency band, but has adifferent smoothness in the frequency direction for a given spatialpoint. Video data has isotropic smoothness within a frame for a fixedtime, but has a different smoothness in the time dimension.

In one embodiment of the present invention, Besov spaces are used toestimate the smoothness in a particular dimension and to determinewhether the multidimensional data is anisotropic. The use of Besovspaces to estimate the smoothness in a particular dimension is describedin detail below. However, the present invention is not limited to theuse of Besov spaces in estimating the smoothness of multidimensionaldata. Any method, algorithm, or numerical calculation that is capable ofestimating data smoothness is suitable for use in connection with thepresent invention.

Besov spaces can be used to measure the smoothness of data. Let the databe represented by a function ƒ:Ω→R, where Ω is an open subset of R^(d),d-dimensional Euclidean space. Assume that there are smoothnessparameters α=(α₁, . . . , α_(d)), with α_(i)>0, and to be specific, thatwe want to measure the smoothness of the data in the space L_(p) (R^(d),so that the data has smoothness α_(i) in the e_(i) direction, i=1, . . ., d, with e_(i)=(0, . . . , 0, 1, 0, . . . , 0) being the ith coordinatevector, which has a 1 as its ith element and 0 as all other elements.

For any h=(h₁, . . . , h_(d)) in R^(d), define the rth difference of ƒin the direction h at the point xεR^(d) recursively asΔ_(h) ^(r)(ƒ,x):=Δ_(h) ^(r−1)(ƒ,x+h)−Δ_(h) ^(r−1)(ƒ,x)andΔ_(h) ⁰(ƒ,x):=ƒ(x)Δ_(h) ^(r)(ƒ,x) is defined on the setΩ_(rh) :={xεΩ|x+khεΩ,k=1, . . . , r}

Let t=(t₁, . . . , t_(d)), t_(i)>0 for all i, and define the rth modulusof smoothness of ƒ in L_(p)(R^(d)) to be${{\omega_{r}\left( {f,t} \right)}_{p}:={{\omega_{r}\left( {f,t,\ldots\quad,t_{d}} \right)}_{p}:={\sup\limits_{{h_{i}} \leq t_{i}}{{\Delta_{h}^{r}\left( {f, \cdot} \right)}}_{L_{p}{(\Omega_{rh})}}}}},{where}$g_(L_(p)(I)) := (∫_(I)f(x)^(p)𝕕x)^(1/p).

The anisotropic Besov space B_(q) ^(α)(L_(p)(Ω)) for 0<α₁, . . . ,α_(d)<r is defined to be the set of all functions ƒ for which${f}_{B_{q}^{\alpha}{({L_{p}{(\Omega)}})}}:=\left( {\sum\limits_{k = 0}^{\infty}\left\lbrack {2^{k}{\omega_{r}\left( {f,2^{{- k}/\alpha_{1}},\ldots\quad,2^{{- k}/\alpha_{d}}} \right)}_{p}} \right\rbrack^{q}} \right)^{1/q}$is finite, and∥ƒ∥_(B) _(q) _(α) _((L) _(p) _((Ω))):=∥ƒ∥_(L) _(p) _((Ω))+|ƒ|_(B) _(q)_(α) _((L) _(p) _((Ω))).Heuristically, a function ƒ in B_(q) ^(α)(L_(p)(Ω)) has α_(i)“derivatives” in L_(p) in the ith coordinate direction.

If α₁= . . . =α_(d), then${f}_{B_{q}^{\alpha}{({L_{p}{(\Omega)}})}}:=\left( {\sum\limits_{k = 0}^{\infty}\left\lbrack {2^{k}{\omega_{r}\left( {f,2^{{- k}/\alpha_{1}},\ldots\quad,2^{{- k}/\alpha_{1}}} \right)}_{p}} \right\rbrack^{q}} \right)^{1/q}$which is equivalent to the usual semi-norm${f}_{B_{q}^{\alpha}{({L_{p}{(\Omega)}})}}:=\left( {\sum\limits_{k = 0}^{\infty}\left\lbrack {2^{\alpha_{1}k}{\omega_{r}\left( {f,2^{k}} \right)}_{p}} \right\rbrack^{q}} \right)^{1/q}$of the isotropic Besov space B_(q) ^(α) ¹ (L_(p)(Ω)) . A more detaileddescription of Besov spaces is provided in Technical Report #328available from the Center for Applied Mathematics at Purdue Universityand is incorporated herein by reference.

In one embodiment of the present invention, multiresolutionrepresentations of data are formed by repeatedly partitioning the datain a first dimension at a first rate, and repeatedly partitioning thedata in a second dimension at a second rate. The first rate is not equalto the second rate. For example, each of the images in the sequence ofimages 201-204 shown in FIG. 2 is a two-dimensional image having a firstdimension 209 and a second dimension 211. The sequence of images 201-204illustrates repeatedly partitioning data in dimension 209 at a firstrate equal to one. The first line 213 in image 201 shows a partitioningof image 201 along the first dimension 209. In the partition illustratedin image 202, each partition in image 201 is divided into two partitionsin dimension 209. In the images 202-204 partition lines 215-228 areadded to illustrate that each of the partitions in each subsequent imageis divided into two partitions. For the sequence of images 201-204, thepartition rate in the first dimension is one, which means thatpartitioning occurs in each image in the sequence of images.

The sequence of images 201-204 also illustrates repeatedly partitioningdata in second dimension 211 at a second rate equal to one-half. Thefirst line 229 in image 202 shows a partitioning of image 201 along thesecond dimension 211. In the partition illustrated in image 202, image201 in the second dimension is divided into two partitions. In theimages 202-204, partition lines 231-232 are added to illustrate thateach of the partitions in alternating subsequent images is divided intotwo partitions.

In an alternate embodiment of the present invention, an estimate ofsmoothness in a first dimension is obtained and an estimate ofsmoothness in a second dimensions is obtained. The smoothness estimatesmay be obtained using Besov spaces, as described above, or any othermethod of estimating smoothness. The first partition rate is set to one,and if the first dimension smoothness estimate is less than the seconddimension smoothness estimate, then the second partition rate is set toa ratio of a first dimension smoothness estimate to a second dimensionsmoothness estimate. If the first dimension smoothness estimate isgreater than the second dimension smoothness estimate, then the secondrate is set to one and the first rate is set to a ratio of the seconddimension smoothness estimate to the first dimension smoothnessestimate.

Multidimensional anisotropic data can be decomposed. If a functionƒ:R^(d)→R is in the anisotropic Besov spaceB_(q) ^(α)(L_(p)(R^(d)))then${f}_{B_{q}^{\alpha}{({L_{p}{(\Omega)}})}}:=\left( {\sum\limits_{k = 0}^{\infty}\left\lbrack {2^{k}{\omega_{r}\left( {f,2^{{- k}/\alpha_{1}},\ldots\quad,2^{{- k}/\alpha_{d}}} \right)}_{p}} \right\rbrack^{q}} \right)^{1/q}$is finite. However, the quantity on the right is equivalent to$\begin{matrix}{{\left( {\sum\limits_{k = 0}^{\infty}\left\lbrack {2^{k/\underset{\_}{\alpha}}{\omega_{r}\left( {f,2^{{- k}\quad{\underset{\_}{\alpha}/\alpha_{1}}},\ldots\quad,2^{{- k}\quad{\underset{\_}{\alpha}/\alpha_{d}}}} \right)}_{p}} \right\rbrack^{q}} \right)^{1/q}\quad{where}}{\underset{\_}{\alpha} = {{\min\limits_{1 \leq i \leq d}{{\alpha_{i}.{Note}}\quad{that}\quad\frac{\underset{\_}{\alpha}}{\alpha_{i}}}} \leq 1}}} & (2)\end{matrix}$with equality only for those i for which α_(i)=α; there is always atleast one i for which this is true. Note also that (2) is equivalent to$\begin{matrix}\left( {\sum\limits_{k = 0}^{\infty}\left\lbrack {2^{k/\underset{\_}{\alpha}}{\omega_{r}\left( {f,2^{- {\lfloor{k\quad{\underset{\_}{\alpha}/\alpha_{1}}}\rfloor}},\ldots\quad,2^{- {\lfloor{k\quad{\underset{\_}{\alpha}/\alpha_{d}}}\rfloor}}} \right)}_{p}} \right\rbrack^{q}} \right)^{1/q} & (3)\end{matrix}$where └y┘ is the greatest integer ≦y, since this increases the size ofeach argument of the modulus of smoothness by at most a factor of 2.

Define S_(k) to be the linear span of the functions${\phi_{j,k}(x)}:={\prod\limits_{i = 1}^{d}{\phi\left( {{2^{\lfloor{k\quad{\underset{\_}{\alpha}/\alpha_{i}}}\rfloor}x_{i}} - j_{i}} \right)}}$for j=(j₁, . . . ,j_(d))εZ^(d). Note that the scaling in each variableis always an integer power of 2, so that S_(k) is, indeed, included inS_(k+1), by the rewrite rule for φ; furthermore, since α/α_(i)=1 for atleast one i, we know that S_(k) is strictly contained in S_(k+1), i.e.,when moving from S_(k) to S_(k+1), one refines functions by a factor oftwo in at least one direction. In fact, going from S_(k) to S_(k+1) werefine in all directions e_(i) for which${\frac{\underset{\_}{\alpha}}{\alpha_{i}}k} < m_{k,i} \leq {\frac{\underset{\_}{\alpha}}{\alpha_{i}}\left( {k + 1} \right)}$for some integer m_(k,i,) and for no other directions.

Thus, if P_(k) is defined to be the projection onto the new S_(k), then${f = {{P_{0}f} + {\sum\limits_{k = 1}^{\infty}\left( {{P_{k}f} - {P_{k - 1}f}} \right)}}},$and P_(k)ƒ−P_(k−1)ƒ is again in S_(k). If there is a function ψassociated with φ and the projections P_(k), then P_(k)ƒ−P_(k−1)ƒ can bewritten as a linear combination of the functions$\prod\limits_{\substack{\begin{matrix}{i \in \Lambda_{k}} \\{\eta_{i} = {{\phi\quad{or}\quad\eta_{i}} = \psi}}\end{matrix} \\ {{not}\quad{all}\quad\eta_{i}} = \phi}}{{\eta_{i}\left( {{2^{\lfloor{{({k - 1})}{\underset{\_}{\alpha}/\alpha_{i}}}\rfloor}x_{i}} - j_{i}} \right)}{\prod\limits_{i \notin \Lambda_{k}}{\phi\left( {{2^{\lfloor{{({k - 1})}{\underset{\_}{\alpha}/\alpha_{i}}}\rfloor}x_{i}} - j_{i}} \right)}}}$for all jεZ^(d), where Λ_(k) consists of the set of coordinates that arerefined in going from S_(k−1) to S_(k). This method can be applied toall orthogonal and biorthogonal wavelets, wavelet frames, or likemultiresolution methods. A more detailed description of multiresolutiondecomposition is provided in Technical Report #328 available from theCenter for Applied Mathematics at Purdue University and is incorporatedherein by reference.

The methods of the present invention may be realized, at least in part,as one or more programs or modules running on a computer—that is, as aprogram or module executed from a computer-readable medium such as amemory by a processor of a computer. The programs are desirably storableon a computer-readable medium such as a floppy disk or a CD-ROM, fordistribution and installation and execution on another (suitablyequipped) computer.

FIG. 3 is a block diagram of one embodiment of a computerized system 300according to the present invention. In one embodiment of the presentinvention, the computerized system 300 is used to form multiresolutionrepresentations of data. The computerized system 300 includes aprocessor 301, a storage medium 303, and a module 305. The processor 301is coupled to the storage medium 303, and module 305 is capable of beingstored on storage medium 303. In one embodiment, the processor 301 is amicroprocessor, however the present invention is not limited to use inconnection with a particular type of processor. Any processor, such as adigital signal processor (DSP), a reduced instruction-set computing(RISC) processor, or a complex instruction-set computing (CISC)processor, capable of processing information is suitable for use inconnection with the present invention. In one embodiment, the storagemedium 303 is a computer-readable storage medium, such as a CD-ROM,floppy disk, or a semiconductor storage device, such as a cache memory.Using a cache memory to store the module permits multiresolutionrepresentations of the data to be quickly generated. The module 305 iscapable of executing on the processor 301 and capable of repeatedlypartitioning data in a first dimension at a first rate, and repeatedlypartitioning data in a second dimension at a second rate, wherein thefirst rate is not equal to the second rate.

In an alternate embodiment of the present invention, data having a timedimension is decomposed using multiresolution decomposition. Exemplarytypes of data having a time dimension include video images, seismicimages, functional (time dependent) magnetic resonance imaging (fMRI)images, and functional (time dependent) positron emission tomography(fPET) images or kinetic positron emission tomography (PET). FunctionalMRI images and functional PET images are generally considered to beimages obtained from techniques involving fast MRI scans, fast PETscans, or techniques for co-registering PET and MRI scans. However, inthe present invention, fMRI and fPET images are any anisotropic imagesobtained using MRI imaging systems, PET imaging systems or combinationsof MRI and PET imaging systems.

In one embodiment, a computer readable medium having computer-executableinstructions for decomposing data having a time dimension includes anumber of operations. First, a spatial dimension smoothness estimate forspatial dimension data is obtained. Second a time dimension smoothnessestimate for the time dimension data is obtained. In one embodiment, thespatial dimension smoothness estimate and the time dimension smoothnessestimate may be obtained by the use of Besov spaces. Third, a first datapartition rate is set to one. Fourth, a second data partition rate isset to a ratio of the time dimension smoothness estimate to the spatialdimension smoothness estimate. Fifth, the spatial dimension data isrepeatedly partitioned at the first rate. Sixth, the time dimension datais repeatedly partitioned at the second rate.

In another alternate embodiment, a method of compressing data having afirst dimension and a second dimension includes forming amultiresolution representation of the data and compressing themultiresolution representation of the data. The method of forming themultiresolution representation of the data includes repeatedlypartitioning the data in the first dimension at a first rate, andrepeatedly partitioning the data in the second dimension at a secondrate, wherein the first rate is not equal to the second rate. Repeatedlypartitioning the data in the second dimension at a second rate comprisesestimating a first smoothness in the first dimension, estimating asecond smoothness in the second dimension, and computing the second rateby forming a ratio of the first smoothness to the second smoothness. Inone embodiment, compressing the multiresolution representation of thedata includes compressing the multiresolution representation of the datausing wavelet compression.

In still another alternate embodiment of the present invention, a methodincludes forming a multiresolution representation of video data having aspatial dimension and a time dimension and processing themultiresolution representation of the video data to remove noise fromthe multiresolution representation of the video data. Processing themultiresolution representation of the video data to remove noise fromthe multiresolution representation of the video data comprises filteringthe multiresolution representation of the video data. In one embodiment,processing the multiresolution representation of the video datacomprises filtering the multiresolution representation of the data usinga low-pass filter. In an alternate embodiment, processing themulti-resolution representation of the video data to remove noise fromthe multi-resolution representation of the video data comprisesfiltering the multiresolution representation of the video data using aband-pass filter. The filters used to process the multiresolutionrepresentation of the video data are typically digital filters.

A method and system for processing multidimensional data has beendescribed. The method and system are based on anisotropicmultidimensional decompositions, including anisotropic waveletdecompositions. The method provides for preparing multidimensionalscalar or vector data with anisotropic smoothness for fuirtherprocessing, such as data compression, noise removal, and reconstruction.

Although specific embodiments have been illustrated and describedherein, it will be appreciated by those of ordinary skill in the artthat any arrangement which is calculated to achieve the same purpose maybe substituted for the specific embodiments shown. This application isintended to cover any adaptations or variations of the presentinvention. Therefore, it is intended that this invention be limited onlyby the following claims and equivalents thereof.

1. A method of forming multi-resolution representations of data, themethod comprising: partitioning the data in a first dimension at a firstrate; and partitioning the data in a second dimension at a second rate,wherein the first rate is not equal to the second rate.
 2. The method ofclaim 1, wherein partitioning the data in a first dimension at a firstrate comprises: partitioning the data in the first dimension at thefirst rate, wherein the first rate is one.
 3. The method of claim 2,wherein partitioning the data in a second dimension at a second rate,wherein the first rate is not equal to the second rate comprises:partitioning the data in the second dimension at the second rate,wherein the second rate is less than one.
 4. A method of compressingdata having a first dimension and a second dimension, the methodcomprising: forming a multi-resolution representation of the data by amethod comprising: partitioning the data in the first dimension at afirst rate; and partitioning the data in the second dimension at asecond rate, wherein the first rate is not equal to the second rate; andcompressing the multi-resolution representation of the data.
 5. Themethod of claim 4, wherein compressing the multi-resolutionrepresentation of the data, comprises: compressing the multi-resolutionrepresentation of the data using wavelet compression.
 6. A computerizedsystem for forming multiresolution representations of data, thecomputerized system comprising: a processor; a storage medium coupled tothe processor; and a module capable of being stored on the storagemedium and capable of executing on the processor, the module beingcapable of partitioning the data in a first dimension at a first rate,and partitioning the data in a second dimension at a second rate,wherein the first rate is not equal to the second rate.
 7. Thecomputerized system of claim 6, wherein the processor is amicroprocessor.
 8. The computerized system of claim 6, wherein theprocessor is a digital signal processor (DSP).
 9. The computerizedsystem of claim 6, wherein the storage medium is a cache memory.